Question

tom determines that the system of equations below has two solutions, one of which is located at the vertex of the parabola. equation 1: (x - 3)² = y - 4 equation 2: y = -x + b in order for toms thinking to be correct, which qualifications must be met? b must equal 7 and a second solution to the system must be located at the point (2, 5). b must equal 1 and a second solution to the system must be located at the point (4, 5). b must equal 7 and a second solution to the system must be located at the point (1, 8). b must equal 1 and a second solution to the system must be located at the point (3, 4).

Explanation:

Step1: Find vertex of parabola

The equation of the parabola is \((x - 3)^2=y - 4\), which can be rewritten as \(y=(x - 3)^2+4\). The vertex form of a parabola is \(y = a(x - h)^2+k\), where \((h,k)\) is the vertex. So the vertex of this parabola is \((3,4)\).

Step2: Substitute vertex into linear equation

The linear equation is \(y=-x + b\). Substitute \(x = 3\) and \(y = 4\) into it: \(4=-3 + b\), so \(b=4 + 3=7\).

Step3: Solve the system of equations

Substitute \(y=-x + 7\) into \((x - 3)^2=y - 4\):

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The solutions for \(x\) are \(x = 2\) and \(x = 3\). When \(x = 2\), \(y=-2 + 7 = 5\); when \(x = 3\), \(y=-3 + 7 = 4\). So the solutions are \((3,4)\) (the vertex) and \((2,5)\).

Answer:

b must equal 7 and a second solution to the system must be located at the point (2, 5).